Conjectured growth series of the three-strand braid group with dual generators

Let $B_3$ be the three-strand braid group, let $\Sigma_3^{\scriptstyle *}$ denote its dual generating set, and let $\mathcal{S}(B_3,\Sigma_3^{\scriptstyle *})$ and $\mathcal{G}(B_3,\Sigma_3^{\scriptstyle *})$ denote the spherical and geodesic growth series, respectively. The conjectured growth-series formulas. The spherical and geodesic growth series are

$$\mathcal{S}(B_3,\Sigma_3^{\scriptstyle *})=\frac{(t+1)(2t^2-1)}{(t-1)(2t-1)^2}, \qquad \mathcal{G}(B_3,\Sigma_3^{\scriptstyle *})=\frac{12t^3-2t^2+3t-1}{(2t-1)(3t-1)(4t-1)}.$$

These rational expressions are conjectured from finite computational data using Padé approximation; the source gives no proof or resolution, so their status remains open.